For summation of n nth roots of unity, we can take summation of all the values of X for n ranging between 0 and n.

Geometrically, all these n nth roots are equally spaced vectors around a unit circle and their sum is nothing but the center of the circle, i.e. 0 + i0

This can also be obtained by integrating X with limit 0 to n.

(hint: remember Integral (sinx )=-cos x and integral (cos x) = sinx; and that sin x = 0 for x =0 or 2 pi. and cos x will be 1 and -1 for 0 and 2 pi and thus sin terms would be 0 and cos terms will cancel out)

Product of n nth roots:

If X^n is the nth root of unity, other roots can be calculated by substituting n = 0 , 1,2...., n.

The product of such a series will be

X^n . X^(n-1)..............X^0 = X^(n +n-1+n-2+......1)

The exponent is an arithmetic series with numbers from 1 to n and summation of such series is given n.n/2